11

Options onOptions onStock Indices, Currencies, Stock Indices, Currencies,

and Futuresand Futures

Chapters 15-16Chapters 15-16

22

European Options on European Options on StocksStocks

Providing a Dividend YieldProviding a Dividend YieldWe get the same probability We get the same probability distribution for the stock price at distribution for the stock price at time time TT in each of the following in each of the following cases:cases:

1.1. The stock starts at price The stock starts at price SS00 and provides a dividend yield = and provides a dividend yield = qq

2.2. The stock starts at price The stock starts at price SS00ee––qq

TT and provides no incomeand provides no income

33

European Options on StocksEuropean Options on StocksProviding Dividend YieldProviding Dividend Yield

continuedcontinued

We can value European options by We can value European options by reducing the stock price to reducing the stock price to SS00ee––qq TT and and then behaving as though there is no then behaving as though there is no dividenddividend

44

Extension of Chapter 9 Extension of Chapter 9 ResultsResults

rTqT KeeSc 0

Lower Bound for calls:

Lower Bound for puts

qTrT eSKep 0

Put Call Parity

qTrT eSpKec 0

55

Example 15.3Example 15.3

SS00 = 300; q = 3%; r = 8%; T = 6 = 300; q = 3%; r = 8%; T = 6 months; K = 290. Find the lower months; K = 290. Find the lower bound for the European Call.bound for the European Call.

C C ≥ 300e≥ 300e-0.03x1/2-0.03x1/2 – 290e – 290e-0.08x1/2-0.08x1/2 = = 16.9016.90

66

ExampleExample

SS00 = 250; q = 4%; r = 6%; T = 3 = 250; q = 4%; r = 6%; T = 3 months; K = 245; c = $10. Find the months; K = 245; c = $10. Find the value of a 3-month European put on value of a 3-month European put on the same index and maturing on the the same index and maturing on the same date as the call.same date as the call.

P=10+245eP=10+245e-0.06x0.25-0.06x0.25 -250e -250e-0.04x0.25-0.04x0.25 = = 3.843.84

77

The Binomial ModelThe Binomial Model

S0u ƒu

S0d ƒd

S0

ƒ

p

(1– p )

f=e-rT[pfu+(1-p)fd ]

88

The Binomial ModelThe Binomial ModelExtention to income-paying underlying Extention to income-paying underlying

assetsassets

In a risk-neutral world the stock price In a risk-neutral world the stock price grows at grows at r-qr-q rather than at rather than at rr when there when there is a dividend yield at the rate of is a dividend yield at the rate of q.q.

The risk-neutral probability, The risk-neutral probability, pp, of an up , of an up movement must therefore satisfymovement must therefore satisfy

pSpS00u+u+((1-p1-p))SS00d=Sd=S00ee ((r-qr-q))TT

so thatso that

pe d

u d

r q T

( )

99

ExampleExample SS00 = 250; q = 4%; r = 6%; T = 3 months; = 250; q = 4%; r = 6%; T = 3 months;

K = 245. Within the next 3 months the K = 245. Within the next 3 months the index can either increase by 10% or drop index can either increase by 10% or drop by 10%. Find the value of the 3-month by 10%. Find the value of the 3-month European put.European put.

f = ef = e-0.06x1/4-0.06x1/4 (1-0.52506)x(245-225) = 9.36 (1-0.52506)x(245-225) = 9.36

52506.09.01.1

9.04/)04.006.0()(

e

du

dep

Tqr

1010

Index OptionsIndex Options Both exchange-traded and OTC Both exchange-traded and OTC Indices: DJX, NDX, RUT, OEX, SPXIndices: DJX, NDX, RUT, OEX, SPX CBOE option contracts are on 100 CBOE option contracts are on 100

times the indextimes the index The most popular underlying The most popular underlying

indices are indices are the Dow Jones Industrial (European) DJXthe Dow Jones Industrial (European) DJX the S&P 100 (American) OEXthe S&P 100 (American) OEX the S&P 500 (European) SPXthe S&P 500 (European) SPX

Contracts are settled in cashContracts are settled in cash

1111

Index Option ExampleIndex Option Example

Consider an American call option on Consider an American call option on an index with a strike price of 560an index with a strike price of 560

Suppose 1 contract is exercised Suppose 1 contract is exercised when the index level is 580when the index level is 580

What is the payoff? If the premium What is the payoff? If the premium is 30, should it be exercised?is 30, should it be exercised?

1212

Using Index Options for Using Index Options for Portfolio InsurancePortfolio Insurance

Suppose the value of the index is Suppose the value of the index is SS00 and the strike and the strike price is price is KK

If a portfolio has a If a portfolio has a of 1.0, the portfolio insurance of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the is obtained by buying 1 put option contract on the index for each 100index for each 100SS00 dollars held dollars held

If the If the is not 1.0, the portfolio manager buys is not 1.0, the portfolio manager buys put options for each 100put options for each 100SS00 dollars held dollars held

In both cases, In both cases, KK is chosen to give the appropriate is chosen to give the appropriate insurance levelinsurance level

1313

Example 1Example 1

Portfolio has a beta of 1.0Portfolio has a beta of 1.0 It is currently worth $500,000It is currently worth $500,000 The index currently stands at 1000The index currently stands at 1000 What trade is necessary to provide What trade is necessary to provide

insurance against the portfolio value insurance against the portfolio value falling below $450,000?falling below $450,000?

1414

Example 1Example 1 Number of puts to buy:Number of puts to buy:

The strike price is 450,000 for 5 The strike price is 450,000 for 5 contracts => 450,000/(100x5) = 900contracts => 450,000/(100x5) = 900

51000*100

000,5001

A

Ph

1515

Example 2Example 2

Portfolio has a beta of 2.0Portfolio has a beta of 2.0 It is currently worth $500,000 and It is currently worth $500,000 and

index stands at 1000index stands at 1000 The risk-free rate is 12% per annumThe risk-free rate is 12% per annum The dividend yield on both the The dividend yield on both the

portfolio and the index is 4% per yearportfolio and the index is 4% per year How many put option contracts should How many put option contracts should

be purchased for portfolio insurance? be purchased for portfolio insurance?

1616

If index rises to 1040, it provides a If index rises to 1040, it provides a 40/1000 or 4% return in 3 months40/1000 or 4% return in 3 months

Total return (with dividends)= 4+1 = Total return (with dividends)= 4+1 = 5%5%

Excess return over risk-free rate=2%Excess return over risk-free rate=2% Excess return for portfolio=4%Excess return for portfolio=4% Increase in Portfolio Value=4+3-1=6%Increase in Portfolio Value=4+3-1=6% Portfolio value=$530,000Portfolio value=$530,000

Calculating Relation Between Calculating Relation Between Index Level and Portfolio Value in Index Level and Portfolio Value in

3 months 3 months

1717

Determining the Strike PriceDetermining the Strike Price

Value of Index in 3 months

Expected Portfolio Value in 3 months ($)

1,080 570,000 1,040 530,000 1,000 490,000 960 450,000 920 410,000

Need to buy 10 put option contractsCost of hedging increases with because more options are

required and each of them is more expensive.

1818

Currency OptionsCurrency Options

Currency options trade on the Currency options trade on the Philadelphia Exchange (PHLX)Philadelphia Exchange (PHLX)

There also exists an active over-the-There also exists an active over-the-counter (OTC) marketcounter (OTC) market

Currency options are used by Currency options are used by corporations to buy insurance when corporations to buy insurance when they have an FX exposurethey have an FX exposure

1919

The Foreign Interest RateThe Foreign Interest Rate

We denote the foreign interest rate by We denote the foreign interest rate by rrff

When a U.S. company buys one unit of When a U.S. company buys one unit of the foreign currency it has an the foreign currency it has an investment of investment of SS00 dollarsdollars

The return from investing at the foreign The return from investing at the foreign rate is rate is rrf f SS00 dollarsdollars

This shows that the foreign currency This shows that the foreign currency provides a “dividend yield” at rate provides a “dividend yield” at rate rrff

2020

Valuing European Currency Valuing European Currency OptionsOptions

A foreign currency is an asset that A foreign currency is an asset that provides a “dividend yield” equal provides a “dividend yield” equal to to rrff

We can use the formula for an We can use the formula for an option on a stock paying a dividend option on a stock paying a dividend yield :yield :

Set Set SS00 = current exchange rate = current exchange rate

Set Set q q = = rrƒƒ

2121

Example 15.4Example 15.4 A foreign currency is currently worth $0.80.A foreign currency is currently worth $0.80. u = 1.02; d = 0.98; T = 2 months; 2 periods u = 1.02; d = 0.98; T = 2 months; 2 periods

(each is 1 month long)(each is 1 month long) R = 6%; rR = 6%; rff = 8% = 8% What is the value of a 2-month European call What is the value of a 2-month European call

option with K = $0.8?option with K = $0.8?

00657.001457.0*45301.0

01457.003232.0*45301.0

0

;03232.0

45301.098.002.1

98.0

12/1*06.0

12/1*06.0

12/1)08.006.0(

eC

eC

CCC

C

ep

u

ddduud

uu

x

2222

Mechanics of Call Futures Mechanics of Call Futures OptionsOptions

They are American style options.They are American style options.

When a call futures option is exercised When a call futures option is exercised the holder acquires the holder acquires

1. A long position in the futures 1. A long position in the futures

2. A cash amount equal to the excess 2. A cash amount equal to the excess of of

the most recent settlement futures the most recent settlement futures price over the strike price price over the strike price

2323

Mechanics of Put Futures Mechanics of Put Futures OptionOption

When a put futures option is When a put futures option is exercised the holder acquires exercised the holder acquires

1. A short position in the futures 1. A short position in the futures

2. A cash amount equal to the excess 2. A cash amount equal to the excess of of

the strike price over the most the strike price over the most recent settlement futures pricerecent settlement futures price

2424

The PayoffsThe Payoffs

If the futures position is closed out If the futures position is closed out immediately:immediately:

Payoff from call = Payoff from call = FF0 0 – – K = K =

=(F=(F-1-1 – K) + (F – K) + (F00 – F – F-1-1))

Payoff from put = Payoff from put = K K – – FF00

where where FF00 is futures price at the time is futures price at the time of the exercise and of the exercise and FF-1-1 is the most recent is the most recent

settlement futures price.settlement futures price.

2525

Put-Call Parity for Futures Put-Call Parity for Futures OptionsOptions

Consider the following two portfolios:Consider the following two portfolios:

1. 1. European call plus European call plus KeKe-rT-rT of cash of cash

2. 2. European put plus long futures European put plus long futures plus plus cash equal to cash equal to FF00ee-rT-rT

They must be worth the same at They must be worth the same at time time TT so thatso that

c+Kec+Ke-rT-rT=p+F=p+F00 e e-rT-rT

2626

ExampleExample European call on silver futures; T = European call on silver futures; T =

6 months; C = $0.56/ounce and K 6 months; C = $0.56/ounce and K = $8.50; F= $8.50; F00 = $8.00 and r = $8.00 and rff = 10%. = 10%.

Find the price of a European put Find the price of a European put with the same T and K.with the same T and K.

Solution:Solution: p = 0.56 + 8.5ep = 0.56 + 8.5e-0.1*0.5-0.1*0.5 – 8e – 8e-0.1*0.5-0.1*0.5 = =

1.041.04 What if p > 1.04?What if p > 1.04?

2727

ExampleExample

Sell the put and the futures, buy Sell the put and the futures, buy the call and invest the PV of K-Fthe call and invest the PV of K-F00

2828

Futures Price = $33Option Price = $4Futures Payoff = $3

Futures Price = $28Option Price = $0Futures Payoff = $-2

Futures price = $30Option Price=?

Binomial Tree ExampleBinomial Tree Example

A 1-month call option on futures has a strike A 1-month call option on futures has a strike price of 29. price of 29.

2929

Consider the Portfolio:Consider the Portfolio: long long futures futuresshort 1 call optionshort 1 call option

Portfolio is riskless when 3Portfolio is riskless when 3– 4 = -2– 4 = -2 or or = 0.8 = 0.8

3– 4

-2

Setting Up a Riskless Setting Up a Riskless PortfolioPortfolio

3030

Valuing the PortfolioValuing the Portfolio( Risk-Free Rate is 6% )( Risk-Free Rate is 6% )

The riskless portfolio is: The riskless portfolio is:

long 0.8 futureslong 0.8 futuresshort 1 call optionshort 1 call option

The value of the portfolio in 1 month The value of the portfolio in 1 month is is -2-2 = -1.6 = -1.6

The value of the portfolio today is The value of the portfolio today is -1.6e -1.6e – 0.06– 0.06 = -1.592= -1.592

3131

Valuing the OptionValuing the Option

The portfolio that is The portfolio that is

long 0.8 futureslong 0.8 futuresshort 1 optionshort 1 option

is worth -1.592 = 0.8is worth -1.592 = 0.8ff - c - c The value of the futures is zeroThe value of the futures is zero The value of the option must The value of the option must

therefore be 1.592therefore be 1.592

3232

Generalization of Binomial Generalization of Binomial Tree ExampleTree Example

A derivative lasts for time A derivative lasts for time TT and is and is dependent on a futures pricedependent on a futures price

F0u ƒu

F0d ƒd

F0

ƒ

3333

GeneralizationGeneralization(continued)(continued)

Consider the portfolio that is long Consider the portfolio that is long futures and futures and short 1 derivativeshort 1 derivative

The portfolio is riskless when The portfolio is riskless when

ƒu df

F u F d0 0

F0u F0 – ƒu

F0d F0– ƒd

3434

GeneralizationGeneralization(continued)(continued)

Value of the portfolio at Value of the portfolio at time time TT is is FF00uu – –FF00 – ƒ – ƒuu

Value of portfolio Value of portfolio todaytoday is is – – ƒƒ

Hence Hence ƒ = – [ƒ = – [FF00uu ––FF00– – ƒƒuu]e]e--rTrT

3535

GeneralizationGeneralization(continued)(continued)

Substituting for Substituting for we obtain we obtain

ƒ = [ ƒ = [ pp ƒ ƒuu + (1 – + (1 – pp )ƒ )ƒdd ]e ]e––rTrT

where where

pd

u d

1

3636

Example 16.4Example 16.4

FF00 = 50. At T=6 months it will be = 50. At T=6 months it will be either 56 or 46. The risk-free rate either 56 or 46. The risk-free rate is 6%. Find the value of a 6-month is 6%. Find the value of a 6-month European call option on the European call option on the futures with a strike price of 50.futures with a strike price of 50.

ƒ = [ 0.4*6 + 0.6*0 ]eƒ = [ 0.4*6 + 0.6*0 ]e–.06*1/2–.06*1/2 = 2.33 = 2.33

4.04656

46501

du

dp

3737

Valuing European Futures Valuing European Futures OptionsOptions

We can use the formula for an option We can use the formula for an option on a stock paying a dividend yieldon a stock paying a dividend yield

Set Set SS00 = current futures price ( = current futures price (FF00))

Set Set qq = domestic risk-free rate ( = domestic risk-free rate (rr ) ) Setting Setting qq = = rr ensures that the expected ensures that the expected

growth of growth of FF in a risk-neutral world is in a risk-neutral world is zerozero

3838

Growth Rates For Futures Growth Rates For Futures PricesPrices

A futures contract requires no initial A futures contract requires no initial investmentinvestment

In a risk-neutral world the expected In a risk-neutral world the expected return should be the risk-free ratereturn should be the risk-free rate

The expected growth rate of the futures The expected growth rate of the futures price is zero in the risk-neutral worldprice is zero in the risk-neutral world

The futures price can therefore be treated The futures price can therefore be treated like a stock paying a dividend yield of like a stock paying a dividend yield of rr

3939

Futures Option Prices vs Futures Option Prices vs Spot Option PricesSpot Option Prices

If futures prices are higher than spot If futures prices are higher than spot prices (normal market), an American prices (normal market), an American call on futures is worth more than a call on futures is worth more than a similar American call on spot. An similar American call on spot. An American put on futures is worth less American put on futures is worth less than a similar American put on spotthan a similar American put on spot

When futures prices are lower than spot When futures prices are lower than spot prices (inverted market) the reverse is prices (inverted market) the reverse is true (see problem 16.12)true (see problem 16.12)

4040

Summary of Key Results Summary of Key Results

We can treat stock indices, We can treat stock indices, currencies, and futures like a stock currencies, and futures like a stock paying a dividend yield of paying a dividend yield of qq For stock indices, For stock indices, qq = average = average

dividend yield on the index over dividend yield on the index over the option lifethe option life

For currencies, For currencies, qq = = rrƒƒ

For futures, For futures, qq = = rr